Can you Cut Copy and Paste like this in Blender?
Yes you can.


Select an object while in object mode
Switch to edit mode Tab
Switch to face select (3 in 2.80)
Select faces that you want to duplicate. Use left click to select in Blender 2.80, right click in Blender 2.79b and earlier (hold SHIFT to select more than one face)
Duplicate selection SHIFT+D
(Optional) Separate the selection into a new object using P > Selection


In object mode you can also duplicate entire objects using SHIFT+D or copy and paste using CTRL+C and CTRL+V.